initial value problem integration

How do I solve this using MATLAB with initial condition h(0)=6 to find the value of t

Answer 1

[Torsten]

You get an implicit equation in h that must be solved numerically for h for a given value of t:

syms h t
f = (-9.5883*h^5+110.5*h^4-340.17*h^3+124.7*h^2+2021.7*h)/sqrt(2*9.81*(h+1));
F = int(f,h,6,h);
G = -pi*0.25^2/4*t;
h50000 = vpasolve(F==subs(G,t,50000),h,6)

h50000 = 

2.828550700405812159269874937196

But finding t for a given value of h is simple:

t50000 = solve(subs(F,h,h50000)==G,t)

t50000 = 

50000.0